Properties

Label 294.6.a.m
Level $294$
Weight $6$
Character orbit 294.a
Self dual yes
Analytic conductor $47.153$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [294,6,Mod(1,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("294.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 294.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.1528430250\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + 66 q^{5} + 36 q^{6} + 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + 9 q^{3} + 16 q^{4} + 66 q^{5} + 36 q^{6} + 64 q^{8} + 81 q^{9} + 264 q^{10} - 60 q^{11} + 144 q^{12} + 658 q^{13} + 594 q^{15} + 256 q^{16} + 414 q^{17} + 324 q^{18} - 956 q^{19} + 1056 q^{20} - 240 q^{22} + 600 q^{23} + 576 q^{24} + 1231 q^{25} + 2632 q^{26} + 729 q^{27} + 5574 q^{29} + 2376 q^{30} + 3592 q^{31} + 1024 q^{32} - 540 q^{33} + 1656 q^{34} + 1296 q^{36} - 8458 q^{37} - 3824 q^{38} + 5922 q^{39} + 4224 q^{40} - 19194 q^{41} + 13316 q^{43} - 960 q^{44} + 5346 q^{45} + 2400 q^{46} + 19680 q^{47} + 2304 q^{48} + 4924 q^{50} + 3726 q^{51} + 10528 q^{52} - 31266 q^{53} + 2916 q^{54} - 3960 q^{55} - 8604 q^{57} + 22296 q^{58} - 26340 q^{59} + 9504 q^{60} + 31090 q^{61} + 14368 q^{62} + 4096 q^{64} + 43428 q^{65} - 2160 q^{66} - 16804 q^{67} + 6624 q^{68} + 5400 q^{69} + 6120 q^{71} + 5184 q^{72} + 25558 q^{73} - 33832 q^{74} + 11079 q^{75} - 15296 q^{76} + 23688 q^{78} + 74408 q^{79} + 16896 q^{80} + 6561 q^{81} - 76776 q^{82} + 6468 q^{83} + 27324 q^{85} + 53264 q^{86} + 50166 q^{87} - 3840 q^{88} + 32742 q^{89} + 21384 q^{90} + 9600 q^{92} + 32328 q^{93} + 78720 q^{94} - 63096 q^{95} + 9216 q^{96} - 166082 q^{97} - 4860 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
4.00000 9.00000 16.0000 66.0000 36.0000 0 64.0000 81.0000 264.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 294.6.a.m 1
3.b odd 2 1 882.6.a.a 1
7.b odd 2 1 6.6.a.a 1
7.c even 3 2 294.6.e.a 2
7.d odd 6 2 294.6.e.g 2
21.c even 2 1 18.6.a.b 1
28.d even 2 1 48.6.a.c 1
35.c odd 2 1 150.6.a.d 1
35.f even 4 2 150.6.c.b 2
56.e even 2 1 192.6.a.g 1
56.h odd 2 1 192.6.a.o 1
63.l odd 6 2 162.6.c.e 2
63.o even 6 2 162.6.c.h 2
77.b even 2 1 726.6.a.a 1
84.h odd 2 1 144.6.a.j 1
91.b odd 2 1 1014.6.a.c 1
105.g even 2 1 450.6.a.m 1
105.k odd 4 2 450.6.c.j 2
112.j even 4 2 768.6.d.p 2
112.l odd 4 2 768.6.d.c 2
168.e odd 2 1 576.6.a.i 1
168.i even 2 1 576.6.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.6.a.a 1 7.b odd 2 1
18.6.a.b 1 21.c even 2 1
48.6.a.c 1 28.d even 2 1
144.6.a.j 1 84.h odd 2 1
150.6.a.d 1 35.c odd 2 1
150.6.c.b 2 35.f even 4 2
162.6.c.e 2 63.l odd 6 2
162.6.c.h 2 63.o even 6 2
192.6.a.g 1 56.e even 2 1
192.6.a.o 1 56.h odd 2 1
294.6.a.m 1 1.a even 1 1 trivial
294.6.e.a 2 7.c even 3 2
294.6.e.g 2 7.d odd 6 2
450.6.a.m 1 105.g even 2 1
450.6.c.j 2 105.k odd 4 2
576.6.a.i 1 168.e odd 2 1
576.6.a.j 1 168.i even 2 1
726.6.a.a 1 77.b even 2 1
768.6.d.c 2 112.l odd 4 2
768.6.d.p 2 112.j even 4 2
882.6.a.a 1 3.b odd 2 1
1014.6.a.c 1 91.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(294))\):

\( T_{5} - 66 \) Copy content Toggle raw display
\( T_{11} + 60 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 4 \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T - 66 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 60 \) Copy content Toggle raw display
$13$ \( T - 658 \) Copy content Toggle raw display
$17$ \( T - 414 \) Copy content Toggle raw display
$19$ \( T + 956 \) Copy content Toggle raw display
$23$ \( T - 600 \) Copy content Toggle raw display
$29$ \( T - 5574 \) Copy content Toggle raw display
$31$ \( T - 3592 \) Copy content Toggle raw display
$37$ \( T + 8458 \) Copy content Toggle raw display
$41$ \( T + 19194 \) Copy content Toggle raw display
$43$ \( T - 13316 \) Copy content Toggle raw display
$47$ \( T - 19680 \) Copy content Toggle raw display
$53$ \( T + 31266 \) Copy content Toggle raw display
$59$ \( T + 26340 \) Copy content Toggle raw display
$61$ \( T - 31090 \) Copy content Toggle raw display
$67$ \( T + 16804 \) Copy content Toggle raw display
$71$ \( T - 6120 \) Copy content Toggle raw display
$73$ \( T - 25558 \) Copy content Toggle raw display
$79$ \( T - 74408 \) Copy content Toggle raw display
$83$ \( T - 6468 \) Copy content Toggle raw display
$89$ \( T - 32742 \) Copy content Toggle raw display
$97$ \( T + 166082 \) Copy content Toggle raw display
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